Testing independence in time series via universal distributions of permutations and words

We study probability distributions of permutations and binary words, which arise in symbolic analysis of time series and their differences. Under the assumptions that the series is stationary and independent we show that these probability distributions are universal and we derive a recursive algorithm for computing the distribution of binary words. This provides a general framework for performing chi square tests of goodness of fit of empirical distributions versus universal ones. We apply these methods to analyze heartbeat time series; in particular, we measure the extent to which atrial fibrillation can be modeled as an independent sequence.